The fitted values, y^=Xβ^, can be written as Hy for an n by n matrix H. The ith diagonal element of H, hi, gives a measure of the influence of the ith value of the independent variables on the fitted regression model. The values of r and the hi are returned by nag_regsn_mult_linear (g02dac).

nag_regsn_std_resid_influence (g02fac) calculates statistics which help to indicate if an observation is extreme and having an undue influence on the fit of the regression model. Two types of standardized residual are calculated:

(a)

The ith residual is standardized by its variance when the estimate of σ2, s2, is calculated from all the data; known as internal studentization.

RIi=ris1-hi.

(b)

The ith residual is standardized by its variance when the estimate of σ2, s-i2 is calculated from the data excluding the ith observation; known as external studentization.

REi=ris-i1-hi=rin-p-1n-p-RIi2.

The two measures of influence are:

(a)

Cook's D

Di=1pREi2hi1-hi

(b)

Atkinson's T

Ti=REin-pphi1-hi.

4 References

Atkinson A C (1981) Two graphical displays for outlying and influential observations in regression Biometrika68 13–20

On entry, the value of a residual is too large for the given value of rms: res[value]=value, rms=value.

7 Accuracy

Accuracy is sufficient for all practical purposes.

8 Parallelism and Performance

Not applicable.

9 Further Comments

None.

10 Example

A set of 24 residuals and hi values from an 11 argument model fitted to the cloud seeding data considered in Cook and Weisberg (1982) are input and the standardized residuals etc calculated and printed for the first 10 observations.